If \(A\subseteq U\), which statement is always true with respect to the universal set \(U\)?
Answer and explanation
Correct answer: \(A\cap A^c=\varnothing\)
The complement \(A^c\) consists of exactly those elements of the universal set \(U\) that are not elements of \(A\). Therefore, no element can belong to both \(A\) and \(A^c\), so their intersection is empty: \(A\cap A^c=\varnothing\). The related identity is \(A\cup A^c=U\), not \(A\). Thus, option A is the only universally valid statement.
Frequently asked questions
What is the correct answer to this question?
\(A\cap A^c=\varnothing\)
Why is this the correct answer?
The complement \(A^c\) consists of exactly those elements of the universal set \(U\) that are not elements of \(A\). Therefore, no element can belong to both \(A\) and \(A^c\), so their intersection is empty: \(A\cap A^c=\varnothing\). The related identity is \(A\cup A^c=U\), not \(A\). Thus, option A is the only universally valid statement.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Complement of a Set and Its Properties.
